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<h1 class="title toc-ignore">Comparison with <code>Knockoff</code>: <span class="math inline">\(\Sigma_X\)</span> is Toeplitz</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-01-18</em></h4>
</div>
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<p><strong>Last updated:</strong> 2018-01-30</p>
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<p><strong>Code version:</strong> 5442ab8</p>
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<section id="introduction" class="level2">
<h2>Introduction</h2>
<p>Applying CASH to linear regression variable selection, compared with other popular methods including BH, Knockoff.</p>
</section>
<section id="simulation-setting" class="level2">
<h2>Simulation Setting</h2>
<p>The simulation setting is very similar to what’s used in Knockoff’s <a href="https://cran.r-project.org/web/packages/knockoff/vignettes/fixed.html">vignette</a>. The only notable difference is that the non-zero signals are normally distributed centered at zero, rather than constant.</p>
<p>Data are simulated by <span class="math display">\[
y_n = X_{n \times p}\beta_p + e_n
\]</span> where <span class="math display">\[
\begin{array}{c}
n = 2000 \\
p = 1000 \\
e_n \sim N(0, 1) \\
\beta \sim \eta\delta_0 + (1 - \eta)N(0, \sigma /\sqrt{n})
\end{array}
\]</span> Each row of <span class="math inline">\(X\)</span> is generated independently from a <span class="math inline">\(N(0, \Sigma_\rho)\)</span> distribution, where <span class="math inline">\(\left(\Sigma_\rho\right)_{j, k} = \rho^{|j - k|}\)</span>, a Toplitz matrix.</p>
<p>Every method selects the variables with respect to a nominal false discovery rate <span class="math inline">\(q = 0.1\)</span>.</p>
<p>In simulations, we are changing the values of the sparsity level <span class="math inline">\(\eta\)</span>, the signal strength <span class="math inline">\(\sigma\)</span>, the feature correlation <span class="math inline">\(\rho\)</span>.</p>
</section>
<section id="methods" class="level2">
<h2>Methods</h2>
<ul>
<li><p>BH: First run multiple linear regression, then apply BH to obtained <span class="math inline">\(p\)</span>-values.</p></li>
<li><p>qvalue: First run multiple linear regression, then apply <code>qvalue::qvalue</code> to obtained <span class="math inline">\(p\)</span>-values.</p></li>
<li><p>Knockoff: Directly apply <code>knockoff::knockoff</code> on <span class="math inline">\(X\)</span>, <span class="math inline">\(y\)</span>.</p></li>
<li><p>ASH: First run multiple linear regression, then apply <code>ashr::ash</code> on obtained <span class="math inline">\(\hat\beta\)</span> and <span class="math inline">\(\hat{\text{se}}\left(\hat \beta\right)\)</span>, using normal mixture and normal likelihood.</p></li>
<li><p>CASH: First run multiple linear regression, then apply <code>cash</code> on obtained <span class="math inline">\(\hat\beta\)</span> and <span class="math inline">\(\hat{\text{se}}\left(\hat \beta\right)\)</span>, using normal mixture and normal likelihood, with default penalty on Gaussian derivative coefficients.</p></li>
<li><p>CASH+: CASH with perfect knowledge, using real noise level <span class="math inline">\(\text{se}\left(\hat{\beta}\right)\)</span>.</p></li>
</ul>
</section>
<section id="observations" class="level2">
<h2>Observations</h2>
<ul>
<li><p>BH is very robust, very fast.</p></li>
<li><p>Knockoff is way too slow and way too conservative with signals being unimodal at zero. Perhaps it needs strong signals distinctly different from the “bulk.” Unimodal setting is really adversary to this method.</p></li>
<li><p><code>CASH</code> works fine, but not better than the basic <code>ASH</code>.</p></li>
</ul>
</section>
<section id="eta-0.9-sigma-5-rho-0.5" class="level2">
<h2><span class="math inline">\(\eta = 0.9\)</span>, <span class="math inline">\(\sigma = 5\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-2-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-2-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Warning in bplt(at[i], wid = width[i], stats = z$stats[, i], out = z$out[z
$group == : Outlier (-Inf) in boxplot 3 is not drawn</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-2-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-0.8-sigma-4-rho-0.5" class="level2">
<h2><span class="math inline">\(\eta = 0.8\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-4-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-4-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Warning in bplt(at[i], wid = width[i], stats = z$stats[, i], out = z$out[z
$group == : Outlier (-Inf) in boxplot 2 is not drawn</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-4-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-0.5-sigma-4-rho-0.5" class="level2">
<h2><span class="math inline">\(\eta = 0.5\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/unnamed-chunk-6-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-in-left0.5-0.6-0.7-0.8-0.9right-sigma-4-rho-0.5" class="level2">
<h2><span class="math inline">\(\eta \in \left\{0.5, 0.6, 0.7, 0.8, 0.9\right\}\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h2>
<section id="overall-across-all-sparsity" class="level3">
<h3>Overall across all sparsity</h3>
<p><img src="figure/knockoff.rmd/avg-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/avg-2.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-0.5-sigma-4-rho-0.5-1" class="level3">
<h3><span class="math inline">\(\eta = 0.5\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/500-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-0.9-sigma-4-rho-0.5" class="level3">
<h3><span class="math inline">\(\eta = 0.9\)</span>, <span class="math inline">\(\sigma = 4\)</span>, <span class="math inline">\(\rho = 0.5\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/100-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
</section>
<section id="eta-in-left0.5-0.6-0.7-0.8-0.9right-sigma-3-rho-0.7" class="level2">
<h2><span class="math inline">\(\eta \in \left\{0.5, 0.6, 0.7, 0.8, 0.9\right\}\)</span>, <span class="math inline">\(\sigma = 3\)</span>, <span class="math inline">\(\rho = 0.7\)</span></h2>
<section id="overall-across-all-sparsity-1" class="level3">
<h3>Overall across all sparsity</h3>
<p><img src="figure/knockoff.rmd/avg%20weak-1.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/avg%20weak-2.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-0.5-sigma-3-rho-0.7" class="level3">
<h3><span class="math inline">\(\eta = 0.5\)</span>, <span class="math inline">\(\sigma = 3\)</span>, <span class="math inline">\(\rho = 0.7\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500%20weak-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/500%20weak-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/500%20weak-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
<section id="eta-0.9-sigma-3-rho-0.7" class="level3">
<h3><span class="math inline">\(\eta = 0.9\)</span>, <span class="math inline">\(\sigma = 3\)</span>, <span class="math inline">\(\rho = 0.7\)</span></h3>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100%20weak-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>No id variables; using all as measure variables</code></pre>
<p><img src="figure/knockoff.rmd/100%20weak-2.png" width="672" style="display: block; margin: auto;" /><img src="figure/knockoff.rmd/100%20weak-3.png" width="672" style="display: block; margin: auto;" /></p>
</section>
</section>
<section id="session-information" class="level2">
<h2>Session information</h2>
<!-- Insert the session information into the document -->
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.2
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ggplot2_2.2.1 reshape2_1.4.3
loaded via a namespace (and not attached):
[1] Rcpp_0.12.14 knitr_1.18 magrittr_1.5 munsell_0.4.3
[5] colorspace_1.3-2 rlang_0.1.6 stringr_1.2.0 plyr_1.8.4
[9] tools_3.4.3 grid_3.4.3 gtable_0.2.0 git2r_0.21.0
[13] htmltools_0.3.6 yaml_2.1.16 lazyeval_0.2.1 rprojroot_1.3-2
[17] digest_0.6.14 tibble_1.4.1 evaluate_0.10.1 rmarkdown_1.8
[21] labeling_0.3 stringi_1.1.6 compiler_3.4.3 pillar_1.0.1
[25] scales_0.5.0 backports_1.1.2 </code></pre>
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